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1、AnalysisofVariance 2 ANalysisOfVAriance ANOVA ANOVAisastatisticalanalysistechniqueinwhichthetotalvariationisdividedintomeaningfulsourcesofvariation RecallfromMeasurementSystemAnalysis 3 AnalysisofVariance ANOVA ANOVAisoftenusedforstudyingtherelationshipbetweenaresponsevariable Y andoneormoreexplanat
2、oryorpredictorvariables X s Thepredictorvariablesarealsocalledfactorsortreatments Whiletheresponseisquantitative thepredictorsmaybeeitherquantitativeorqualitative However quantitativepredictorsareanalyzedasiftheyarequalitative orcategorical 4 AnalysisofVariance ANOVA Tostudytheeffectofoneormorefacto
3、rsonaresponse eachfactorhavingtwoormorelevels ANOVAcanbeusedtodeterminethestatisticalsignificanceofeffectscalculatethecomponentsofvarianceestimatethecontributiontovariationbyeachidentifiedsourceestimatetheunderlyingnoisewithintheprocess 5 ANOVA Principle N 1 1 N 2 2 N 3 3 N 4 4 6 ANOVA Terminology y
4、ijisthejthobservationwhenthefactorisatitsithlevel y14 y23 y31 y42 7 ANOVA Terminology yi istheexpectedresponsewhenthefactorisatitsithlevel 8 ANOVA Terminology y istheoverallmeanofallobservationsacrossallfactorlevels 9 ANOVA Terminology i yi y istheeffectofthefactoratitsithlevel 10 ANOVA Terminology
5、ij yij yi istheresidualorerrorofthejthobservationattheithlevelofthefactor 11 ANOVA Terminology TotalDeviation ExplainedDeviation UnexplainedDeviation 12 ANOVAtable 1 DFTotal DFFactor DFErrorDFError DFTotal DFFactor N 1 k 1 N k2 SSTotal SSFactor SSError 13 ANOVATable TotalDeviation ExplainedDeviation
6、 UnexplainedDeviation SSTotal SSExplained SSUnexplained SSFactor SSError 14 HypothesisTesting H0 1 2 kallgroupmeansareequalHa i jforsomei jatleastonepairofgroupmeansisnotequalANOVAverifiesthenullhypothesisbycomparingthevariancebetweenthegroupsagainstthevariationwithinagroupmean Thenullhypothesisisre
7、jectedif 15 HypothesisTesting Ingeneral Aswillbeseeninfollowingexamples theactualformulationofF isdependentonthetypeofANOVAmodel 16 ANOVA Assumptions Theobservationsaremutuallyindependent Stat Nonparametrics RunsTestThekgroupsexhibithomogeneityofvariance i e 1 2 k Stat ANOVA TestforEqualVariancesThe
8、residualsarenormallydistributed i e e NID 0 Stat BasicStatistics NormalityTest 17 ANOVA Assumptions Independance Homoscedasticity Normality 18 ANOVA Models FixedEffects ANOVAI Modelfactorlevelsintheexperimentarespecificallychosentheconclusionisrelevantonlyforthechosenlevelsandmaynotbeextendedtosimil
9、artreatmentsthatwerenotconsideredRandomEffects ANOVAII ModelfactorlevelsarerandomlydrawnfromalargepopulationoftreatmentstheconclusionisvalidfortheentirepopulationoftreatmentsMixedEffects ANOVAIII Modeloneormorefactorsarefixedfactorswhileothersarerandomfactors 19 ANOVA Models Ak WayANOVAmodelinvolves
10、thestudyofkfactors k WayANOVAI kfactors allfactorsoffixedeffectk WayANOVAII kfactors allfactorsofrandomeffect WillnotbediscussedinGBcourse k WayANOVAIII kfactors somefactorsoffixedeffect WillnotbediscussedinGBcourse othersofrandomeffect 20 One WayANOVA IModel whereYijisthevalueoftheresponsevariablei
11、nthejthtrialfortheithfactorlevelortreatment istheoverallmeanforallobservations i i istheeffectoftheithtreatments t ijistherandomerrorcomponents t Thehypothesismaynowberephrased H0 1 2 r 0Ha notall iequal0 21 One WayANOVA IModel 22 Example Anengineerwanttoevaluatetheeffectofthepercentageofthestrength
12、eneroftheadhesivetothebondingforce breakageforceoftheassy ofabondedactuator Theengineerdecidestoinvestigatefourlevelsofpercentageofthestrengthener ie 5 10 15 20 Sixsamplesweretestedateachpercentagelevel Thedataisstoredin 4 ANOVA MTW 23 Example Stat ANOVA One way 24 Example SessionWindowsOne wayANOVA
13、 ForceversusStrengtherAnalysisofVarianceforForceSourceDFSSMSFPStrength3382 79127 6019 610 000Error20130 176 51Total23512 96Individual95 CIsForMeanBasedonPooledStDevLevelNMeanStDev 5610 0002 828 10615 6672 805 15617 0001 789 20621 1672 639 PooledStDev 2 55110 015 020 025 0 25 Example 26 Example 27 Ex
14、ample Stat Regression ResidualPlots 28 Example 29 One WayANOVA IIModel whereYijisthevalueoftheresponsevariableinthejthtrialfortheithfactorlevelortreatment istheoverallmeanforallobservations i i istheeffectoftheithfactorlevels t ijistherandomerrorcomponents t Thehypothesismaynowberephrased H0 0Ha 0 3
15、0 One WayANOVA IIModel ThereisnodifferenceincalculationoftheF StatisticbetweenaOne WayANOVA IModelandaOne WayANOVA IIModel Thedifferenceliesintheconclusion 31 Example Anengineerisinterestedinwindingmachinevariabilityinpeelstrengthofacoil Theengineerselectsfourwindingmachinesatrandomanddeterminesthep
16、eelstrengthoffoursampleschosenatrandomfromeachwindingmachine 32 Example Stat ANOVA GeneralLinearModel 33 Example SessionWindowsGeneralLinearModel PeelStrengthversusMachineFactorTypeLevelsValuesMachinerandom41234AnalysisofVarianceforPeelStr usingAdjustedSSforTestsSourceDFSeqSSAdjSSAdjMSFPMachine389 18789 18729 72915 680 000Error1222 75022 7501 896Total15111 937 34 Example EndofPresentation Rev125Jun02
